The Last Supper was indeed a Mass. It was the anticipated sacrifice that would happen on Good Friday.
As to the issue of which way Jesus faced...it wouldn't matter simply for the fact that He is God, and so He could face whatever He wants. This kind of questioning sort of reduces to speaking about Christ in purely human terms, which when carried to its logical conclusion, gives rise to those who seek only the "historical Jesus" - aka Arianism.
Since the Church was not in existence until Pentecost, the Last Supper could not have been a Mass.
The Eucharist was instituted, yes. But a Mass was not said. Recall that the Mass is the re-presentation of the Sacrifice of Calvary.